15

In addition to the variational approach based on Noether's theorem, there are other ways to find conservation laws for nonlinear PDE's: The symmetry/adjoint symmetry pair method extracts a conservation law from a bilinear skew-symmetric identity. It involves the following steps: (a) Linearize the given system of PDEs; (b) Find the adjoint system of the ...


9

First a comment: in the context of nonlinear wave and Klein-Gordon equations, the venerable "ABC method" of Cathleen Morawetz is literally "mucking around until you see something". (The A, B, and C refer to the three free function coefficients you can choose for your argument and judicious choices give you good signs for the things you ...


6

Another construction when your PDE has a Hamiltonian structure, but the Poisson structure has a non-trivial kernel. Roughly speaking, this means that the phase space is foliated by submanifolds, and each of these varieties is equiped with a non-degenerate Poisson structure, this one varying smoothly along the foliation. Functions that are constant along each ...


6

One way you can use conservation laws of a PDE is in numerics; you check at each moment in time the value of the conserved quantity coming from the conservation law, to see if it is still being conserved approximately, and as soon as it is clearly not conserved (and not nearly conserved), you should no longer believe that the numerical approximation to the ...


4

When $n=1$, you can always do this, at least near $t=0$, by solving a single inhomogeneous, linear first-order PDE; you can even arrange that $h_2 = h_1$. When $n>1$, there is a geometrical obstruction that can be computed in terms of $f_1$ and $\lambda g$. This is a classical fact in the geometry of PDE and characteristic systems. Here is a summary of ...


3

Another application for higher symmetries is (hypothetical) sufficient condition for integrability. Existence of infinitely many genuinely higher (i.e. other than point or contact and of order greater than one) symmetries is generally believed to imply that the system under study either can be transformed into a linear one (cf. e.g. the Burgers equation) ...


3

Carlo's answer brings partial information. Let me complete it. First of all, a general gas is not polytropic, thus the pressure is not proportional to $e-\frac12\rho u^2=:\rho\varepsilon$. In other words, there is not such beast as $\gamma$, or it is not constant. Thus the entropy $\eta$ is not given by a log-algebraic formula in terms of $p$ and $\rho$. ...


2

This should mean that $U_{\mu_k} \to U$ almost everywhere on $\mathbb{R}^m \times [0,T)$, and moreover the sequence of functions $U_{\mu_k}$ is uniformly bounded: $$\sup_k \sup_{(x,t) \in \mathbb{R}^m \times [0,T)} |U_{\mu_k}(x,t)| < \infty.$$ I suppose that the functions $U_{\mu_k}$ take their values in $\mathbb{R}$ or $\mathbb{C}$ or some other obvious ...


2

Yes, the mathematical entropy is more general than the physical entropy, but in many contexts of physical relevance these turn out to be the same quantities. In particular, for the one-dimensional Euler equation the conserved quantities $U$ are the mass density $\rho$, momentum density $\rho u$ (with velocity $u$), and the energy density $e$. The ...


2

The fact that integrable evolutionary partial differential systems have infinitely many integrals of motion (conserved quantities) is indeed of (differenial-)algebraic nature and is not related to the choice of boundary conditions. In the KdV case this can be established in a very simple fashion using the (generalized) Miura transformation, see e.g. here. ...


2

If there is no connection between $\rho$ and $j$, you may as well combine them into a 4d vector $F=(\rho,j)$. If I understand it correctly, the question is: If $F$ is square integrable and divergence-free, can it be approximated by divergence-free vector fields of compact support? This is true and can be shown along the following lines: First define an ...


2

Here's one possible solution, but this may or may not be what your professor had in mind. Since $\lambda$ is a constant, we can ignore it by absorbing it into $g$. Assume $u$ is scalar (takes value in $\mathbb{R}^1$). Assume that $g$ is a function of $u$ only. Let $G(s; u_0)$ be the solution to $\partial_s G = g(G)$ with initial data $G(0;u_0) = u_0$. ...


2

Writing your equation (3) as $a(u,\psi)=0$, it is indeed common to call $u\in L^1_\mathrm{loc}$ a `weak solution' to your problem if and only it satisfies $$ a(u,\psi) = 0 \mbox{ for all } \psi \in C_c^\infty(\mathbb{R}\times [0,T)) $$ However, this is a definition which implicitly encodes $C^\infty_c(\mathbb{R}\times [0,T))$ as the base set of 'test ...


2

You may find it helpful to think of the area integral $\int f(x)dx $ as the zero-frequency component $\hat{f}(0)$ of the Fourier transform $\hat{f}(\omega)$. The statement "the area of a convolution equals the product of areas" is then the special $\omega\rightarrow 0$ case of the more general statement that the frequency-$\omega$ component of a ...


2

I am aware of some real-world applications which I learnt from Chapter 1 of "Hyperbolic Partial Differential Equations. Theory, Numerics and Applications" by Meister and Struckmeier. This chapter presents plenty of scenarios which can me modelled using balance laws, which can be reduced to hyperbolic conservation laws when the model is simpliefied ...


2

Some of them: Multiply by $u_x$ and integrate, you get $$ \int u_{tt} u_x ~dx = 0 $$ so $$ \partial_t \int u_{t} u_x ~dx - \int u_t u_{tx} ~dx = 0 $$ the second term integrates to zero. Multiply by $u_t$ and integrate by parts you get $$ \int u_{tt} u_t + (1 + \int u_x^2 ~dx) u_{xt} u_x + uu_t - u^{2r+1} u_t ~dx = 0 $$ This you can rewrite as $$ \...


1

More (too long) a comment than an answer. It is somewhat paradoxical that almost 90 years after Serguei Sobolev introduced the notion of weak solutions of PDE, also almost 70 years after Laurent Schwartz wrote his treatise on Distribution theory, we keep writing weak solutions using test functions. For the example of the IVP for the Incompressible Navier-...


1

No. For example, suppose $x_0 \in \mathbb{R}^N$ is an asymptotically stable equilibrium point of the gradient flow; suppose $- \nabla L (x_0) = 0 $ and the matrix $- \nabla^2 L (x_0) $ has $N$ negative eigenvalues. Then there exists an open set $U$ about $x_0$ such that every point $y \in U$ limits to $x_0$ in forward time. If there is some continuous ...


1

Some of nice applications of conservation laws that are not mentioned a lot are in the electrophoresis and hromatography. A good place to start to learn about those two are books: Babskii, Zhukov, Yudovich, Mathematical Theory of Electrophoresis, 1989 Rhee, Aris, Amundson, First order partial differential equations, 1989 Also, the classical books with a ...


1

Yes, this is a classical approach in nonlinear PDEs. Here is one example: Sometimes the starting and suitable notion of weak solution is so weak that one can prove existence but not uniqueness. The typical statement then becomes: if, among these possibly many weak solutions, there exists a strong one, then it is the unique only (weak) one. The real problem ...


1

I'm not sure this really qualifies as an answer (there's not much here beyond notation really), but one general way of thinking about this sort of problem is as follows. Sorry if any of this seems a bit basic; I am in no way an expert in this area. All I'm going for here is a fleshing out of the general shape of how I prefer to regard this sort of problem. (...


1

If $j$ has compact support in $\mathbb R_t\times\mathbb R^3_x$, $\rho$ will have compact support iff $\int_{-\infty}^\infty\nabla\cdot j_s\ ds\equiv0$. The set of approximable $L^2$ solutions should be the closure of these.


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